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DIAGRAMS 


FOR  THE  SOLUTION  OF  THE 


KUTTER  AND   BAZIN  FORMULAE 


FOR  THE 


FLOW  OF  WATER 


When  any  three  of  the  four  variables,  Velocity. 
Slope,  Hydraulic  Radius,  and  Roughness,  are  known, 
the  fourth  can  be  read  off  at  once,  in  English  or  metric 
units,  without  using  a  straight  edge. 


PREPARED  BY 

KARL  R.  KENNISON,  M.  ME.  Soc.  C.  E. 

815  Grosvenor  Building 

PROVIDENCE,  R.  I. 

1913 


PRICE  $1.00 


/v 

\ 


THE  KUTTER  AND  BAZIN  FORMULAE 

In  the  absence  of  actual  discharge  measurements,  which  should  always  be  preferred 
to  the  best  computations,  the  formulae  represented  in  these  diagrams  are  commonly  relied 
on  to  compute  the  velocity  of  water  flowing  in  open  channels  and  pressure  conduits.  The 
Kutter  formula  is  generally  preferred  to  the  Bazin  formula,  for  all  classes  of  channels, 
especially  hi  computations  of  river  flow.  The  Bazin  formula  is  not  generally  applied  to 
channels  over  twenty  feet  wide. 

These  formulae  assume  a  condition  of  uniform  flow,  and  depend  for  accurate  results 
on  the  right  choice  of  a  coefficient  of  roughness  to  fit  the  channel  hi  question.  Reference 
should  be  had  to  the  many  published  works  on  Hydraulics  for  a  discussion  of  the  proper 
application  of  the  formulae,  the  measurement  of  surface  slope,  the  determination  of  the 
coefficient  of  roughness,  and  the  effects  of  bends  and  irregularities  in  the  channel  bed, 
which  practically  increase  the  coefficient  of  roughness.  Uncertainties  in  the  application 
of  the  formulae  do  not  warrant  a  more  precise  solution  than  can  be  obtained  easily  with 
these  diagrams. 

NOTATION 

V....Mean  velocity  of  water  in  uniform  motion,  in  feet  per  second.  (The  marginal 
scale  of  velocities  is  in  meters  per  second.) 

s.... Slope  of  free  water  surface  or  hydraulic  gradient,  friction  head -r  length. 

S....1000  x  s,  or  slope  in  feet  per  thousand  feet  (or  meters  per  thousand  meters). 

R.  ..Hydraulic  radius,  or  sectional  area  of  stream-i-wet  perimeter,  in  feet.  (The 
marginal  scale  of  hyd.  radii  is  in  meters.) 

For  ordinary  river  beds,  R— practically  the  mean  depth. 

n.... Coefficient  of  roughness  in  Kutter  formula. 

y.... Coefficient  of  roughness  in  Bazin  formula. 

Some  of  the  values  of  n  and  y  in  common  use  are  shown  below  the  diagrams.  They 
are  average  values  and  should  be  varied  to  suit  the  condition  of  the  surface  in  question: 
e.  g.,  for  planed  boards  well  laid  and  with  smooth  end  joints  n  is  commonly  assumed 
=.009  instead  of  .010:  For  concrete  lined  tunnels,  where  only  ordinary  care  is  taken  to  obtain 
a  smooth  interior  and  where  the  obstruction  due  to  vegetable  growths  must  be  anticipated, 
n  should  be  assumed=.013  or  .014  instead  of  .010,  the  value  given  for  smooth  cement: 
Swollen  rivers  encumbered  with  detritus  might  require  a  value  of  n  as  high  as  .045,  or  even 
higher  in  torrents  spending  part  of  their  energy  in  rolling  boulders  along  and  across  the 
bottom. 


288611 


EXAMPLES  ILLUSTRATING  USE  OF  DIAGRAMS 

(1)  To  find  the  carrying  capacity  of  a  circular  tunnel  8  ft.  in  dia.,  lined  with  brick- 
work in  poor  condition,  flowing  full  under  a  hydraulic  gradient  of  5  feet  per  mile. 

The  roughness  of  this  lining  is  about  equal  to  that  of  rubble  masonry,  or  say  n=.017. 
R=2  ft.  S=.95  ft.  per  thousand  ft.  Area  of  tunnel=50.3  sq.  ft.  A  vertical  line  up  from 
the  intersection  of  n=.017  and  R=2  intersects  S=. 95  at  V=4.3.  Therefore  the  carrying 
capacity— 4.3  ft.  per  sec.  x  50.3  sq.  ft.=216  cu.  ft.  per  sec. 

Practically  the  same  result  would  have  been  obtained  by  the  Bazin  diagram,  using 
y=.46. 

(2)  The  following  simultaneous  stream  measurements  were  made  covering  a  certain 
river  length:    Di-charge=3340  cu.  ft.  per  sec.,  S=2.35  ft.  per  thousand  ft.,  mean  V=5.2 
ft.  per  sec.,  mean  R=4.0  ft.    From  the  computed  coefficient  of  roughness,  which  under  the 
conditions  of  this  problem  we  may  assume  constant,  find  the  slope  along  this  stretch  of 
the  river,  when  the  same  quantity  is  flowing,  obstructed  by  a  dam  at  some  point  down- 
stream which  raises  the  water  surface  so  that  mean  R— 12.5  ft.  and  mean  sectional  area 
of  stream=2350  sq.  ft. 

A  vertical  line  down  from  the  intersection  of  V=5.2  and  S=2.35  intersects  R=4.0 
at  n=.035. 

After  the  dam  is  built,  the  velocity  is  3340  cu.  ft.  per  sec.-r-2350  sq.  ft.=  1.42  ft.  per 
sec.  A  vertical  line  up  from  the  intersection  of  n=.03o  and  R=12.5  intersects  V=1.42 
at  a  point  lying  in  the  group  of  lines  for  S=.03  but  below  the  one  corresponding  to  n=.035. 
Interpolating  therefore  between  the  lines  S=.01  and  S=.03,  using  in  each  case  the  one 
corresponding  to  n=.035,  we  find  that  our  point  lies  on  S=.025,  which  is  the  required 
slope  in  ft.  per  thousand  ft.  due  friction  V\eai 

This  last  example  illustrates  the  application  of  the  diagrams  to  the  problem  of  "back- 
flowage"  above  dams  and  other  obstructions. 


The  intersection   of 

V  (Mean  Velocity   in  Ft.  per  sec.) 

and  S  (Slope    in  ft.  per  thousand  ft.) 

is  vertically  over 

the  intersection  of 

n  (Coefficient  of  Roughness 
and  R  (Hydraulic  Radius  in  ft.) 


Note  that  S  in  diagram 
1000  X  s  in  Formula 


V=  1.2- 


v= 


For    Metric    Units 

7o  .    I    .    .00155 

"+^rr+  — s~~ 


-.6=R 


Diagram 

FLOW  OF  WATER 

Formula  of  Ganguillet  and  Kutter 
commonly  called  the 

KUTTER_FORMULA 

By    Kar~l    R.  Kennison 

C-Opyright      1312. 
For    English   Units 

41.6 


Some  values  of  n  in  common  use:  *  Planed  boards  or  smooth  cement,  .010;      Well  laid  brickwork,  .013;        Rubble  masonry,  .017;        Very  firm  gravel,  .020; 
Earthen  canals  in  good  order,  .025;        Ordinary  earthen  river  beds  with  occasional  stones  and  weeds,  .030;        Earthen  river  beds  in  bad  order,  .035  or  more. 


The  intersection   of 
V  (Mean  Velocity   m  ft-  per  se 
and  S  (Slope   in  ft.  per  thousanc 
is  vertically  over 
the  intersection   oF 
Y  (Coefficient  of  Roughness) 
and  R  (Hydraulic  Radios  in  ft.) 

V. 

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Some  values  of  y  in  common  use:     Planed  boards  or  smooth  cement,  .06;       Well  laid  brickwork,  .16;      Rubble  masonry,  .46;      Very   firm    gravel.    .80; 
Earthen  canals  in  good  order,  1.30;  in  bad  order,   1.75.      For  computing  river  flow  the  Kutter  formula  is  generally  considered  more  accurate  than    Baiin. 


Kemington  Press,  Providence 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


I 


